Timeline for Approximation of topological dynamical systems?
Current License: CC BY-SA 3.0
7 events
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| Aug 24, 2015 at 11:11 | comment | added | Lasse Rempe | Another situation that comes to mind is the convergence of families of unicritical polynomials, parameterised as $z\mapsto (1+z/d)^d+c$, to the exponential family $z\mapsto e^z+c$. The convergence seems "dynamical" in a certain sense. There is an old preprint by Devaney-Goldberg-Hubbard, now published in two parts with additional co-authors. Also, additional results in a similar spirit by Kriete, Krauskopf and others; see arxiv.org/abs/0910.0743 by my former student Helena Mihaljević-Brandt. However, I am not sure this "dynamical convergence" has ever been fully formalised as a concept. | |
| Aug 21, 2015 at 13:55 | comment | added | Giraffro | This seems like the right concept, although I'm probably asking for something weaker than being 'locally topologically conjugate'. But this is a nice direction to look in. Thank you! | |
| Aug 21, 2015 at 13:42 | comment | added | Lasse Rempe | Of course, there are situations where you may not have structural stability, but still have "dynamical convergence" along certain sequence. For example, as pointed out below, the Julia set of a quadratic polynomial is not continuous at c=1/4. However, if you approach through the interval $c\in (0,1/4)$, then the Julia sets will converge (and the maps are even topologically conjugate on the Julia set - where they are topologically just the doubling map on the circle). | |
| Aug 21, 2015 at 13:39 | comment | added | Lasse Rempe | You may be looking for the notion of Structural Stability: en.wikipedia.org/wiki/Structural_stability | |
| Aug 19, 2015 at 15:49 | answer | added | Dylan Thurston | timeline score: 6 | |
| Aug 19, 2015 at 11:57 | review | First posts | |||
| Aug 19, 2015 at 12:39 | |||||
| Aug 19, 2015 at 11:53 | history | asked | Giraffro | CC BY-SA 3.0 |